Sunday, August 28, 2011

The Cutting of the Traveler's and Prisoner's Gordian Knot

I've observed a widespread impression that the traveler's dilemma is not solved as rationally as the arguments make it seem. Similarly, the prisoner's dilemma. (Inspiration.)

I've worked out that this intuition is correct. The solution really is to cooperate. The reason nobody thinks of that is that we can imagine a real-world situation, where the solution is deviate, and we can imagine a pure rationality situation only imperfectly...most of the time.

From the inspiration: "Assume all the usual ridiculous things: common knowledge of rationality, risk neutrality, pure selfishness, etc."

Common knowledge of rationality is the key. You're both familiar with the Nash argument to deviate, but this is a failure of meta-strategy. If you want to maximize your own payoff, you need to pick the strategy with the highest Nash equilibrium. If you're both rational and know the other's rational, you'll both pick the same strategy.

The strategy with the highest payoff Nash equilibrium is to cooperate. (Further, this strategy is fairly robust against failures of rationality.)

In essence, the classical solution assumes deviation when it says that choosing $99 'dominates' $100. Actually, choosing the deviation strategy equilibrates at $2, and choosing cooperate equilibrates at $100, and so choosing $99 implies a strategy that is worse than the strategy that chooses $100. Going further, the problem was that they stopped when they found a solution. Ratiocination requires you to keep going, to work out further consequences, mainly to check for contradictions. Lo and behold, an answer is not the answer.

I idly wonder if this is related to the axiom of choice in set theory, or if that axiom is ludicrously badly named.

Update: "What’s funny is that that’s so hyperrational that it’s insanely and literally idiotic. An actual person could never possibly do that." An actual hyper-rationality wouldn't do it either. This is probably a case of intuition sneaking in the back door. Update: apparently he agrees with the $100 strategy but doesn't fully understand why it's optimal.